Abstract:

Let $ E $ be a Moran set on $ {{\Bbb R}} ^{1} $ associated with a bounded closed interval $ J $ and two sequences $ (n_{k})^{\propto}_{k=1} $ and $ {\cal C}_{k}=((c_{k, j})_{j=1}^{n_{k}})_{k\geq 1} $ of numbers. Let $ \mu $ be the Moran measure on $ E $ determined by a sequence $ ({\cal P}_{k})_{k\geq1} $ of positive probability vectors. For every $ n\geq 1 $, let $ C_{n}(\mu) $ denote the collection of all the $ n $-optimal sets for $ \mu $ with respect to the geometric mean error; let $ \alpha_n\in C_n(\mu) $ and $ \{P_{a}(\alpha_{n})\}_{a\in\alpha_{n}} $ be an arbitrary Voronoi partition with respect to $ \alpha_n $. We prove that $ \min\limits_{a\in\alpha_n}\mu (P_{a}(\alpha_{n})), \max\limits_{a\in\alpha_n}\mu (P_{a}(\alpha_{n}))\asymp n^{-1}. $ For each $ a\in\alpha_{n} $, we show that the set $ P_{a}(\alpha_{n}) $ contains a closed interval of radius $ d_{2}|P_{a}(\alpha_{n})\cap E| $ which is centered at $ a $, where $ d_{2} $ is a constant and $ |B| $ denotes the diameter of a set $ B\subset{{\Bbb R}} ^1 $. Let $ e_n(\mu) $ denote the $ n $th geometric mean error for $ \mu $ and $ \hat{e}_n(\mu):=\log e_n(\mu) $. We show that $ \hat{e}_n(\mu)-\hat{e}_{n+1}(\mu)\asymp n^{-1} $.

Key words: Geometric mean error, Optimal Voronoi partition, Moran measure